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MCA 520: Graph Theory Instructor Neelima Gupta [email protected] Table of Contents Walks, Trails and Paths Walks • May have repeated Edges and Vertices. • In case of multi-graph, we include the edges also. In a simple graph, we can omit the edges and simply mention the sequence of vertices. • Closed Walk Trails • No repeated edges but vertices may repeat. • Closed Trail Path • Neither vertices nor edges repeat. • Definition: – we say that a u-v walk W contains a u-v path P if all the edges and vertices of P occur in W and in that order but not necessarily consecutive. – Similarly a closed walk W contains a cycle C if ….. • Lemma: Every u-v walk contains a u-v path Odd/Even Walk • Odd/Even walk : number of edges is odd/even • Lemma: Every closed odd walk contains an odd cycle. • Remark: A closed even walk need not even contain a cycle, it may simply repeat edges. But, if an edge e appears exactly once in a closed walk, then the walk contains a cycle through e. Even Graph • A vertex is stb even(/odd) if its degree is even(/odd). • A graph is stb an even graph if all its vertices are even. Maximal Path • A path in a graph is stb maximal if it is not contained in a longer path. – If a graph is finite, maximal path always exists. • If every vertex in a finite graph G has degree at least 2 then it contains a cycle. – This is not true if the graph is not finite. Connection Relation • (u,v): u is stb connected to v …… • Symmetric, Reflexive, Transitive • Equivalence Relation • Equivalence Class: Connected Component • Lemma: A graph with n vertices and k edges has at least n – k components. • Proof: A graph with no edges has n components. Adding an edges reduces the number of components by at most 1. Thus after adding k edges, number of components is at least n – k. Deleting an edge/vertex • G – e: Deleting an edge does not delete its incident vertices. • G – v: Deleting a vertex delete its incident edges. • Thus deleting an edge may increase the number of components by at most 1. • Deleting a vertex v may increase the number of components by (more) at most deg(v) – 1. • Induced Graph G[T] = Graph that remains after deleting some vertices such that the set of remaining vertices is T. i.e. G[T] = (T, E(T)), where E(T) = {(u, v):u,v are in T and (u,v) is an edge in G} • Every subgraph of a graph need not be an induced subgraph. Cut-edge and Cut-Vertex • An edge e is stb a cut edge if … • A vertex v is stb a cut vertex if … Characterize cut-edges in terms of cycles. • Theorem: An edge is a cut edge iff it does not belong to any cycle. Bi-partite Graphs • Konig Theorem : Characterizing Bipartite Graphs in terms of cycles: A graph is bipartite iff it has no odd cycles. • Testing whether a graph is bipartite: Union of Graphs • Definition: • K4 : a union of two 4-cycles. • Kn can be expressed as a union of k bi-partite graphs iff n < 2k. Eulerian Circuits • A graph is Eulerian if it has a closed trail containing all the edges. • A graph is Eulerian iff it has at most one nontrivial component and all its vertices have even degree.